To illustrate the use of our method for a more complex scenario, we consider a beam oscillating at high amplitudes compared to its length. The non-negligible rotational deformation invalidates the linear strain assumption. As a consequence, a geometrically non-linear model needs to be considered. In the domain \(\Omega \)  of the reference configuration, the variational statement governing free vibration reads [1]}, [2]}:
\(&\text{Find }\mbox{{$u$}}\in \mbox{{$H$}}^1\big (\mathcal {T},\mbox{{$H$}}^1_{0}(\Omega )\big ) \text{ s.t., for a.e. }t,\,\forall \,\mbox{{$v$}} \in \mbox{{$H$}}^1_{0}(\Omega ):\nonumber \\& \quad \big \langle \rho _0\frac{\partial ^2 }{\partial t^2}\mbox{{$u$}} , \mbox{{$v$}} \big \rangle _{H^{-1},H^{1}_0} + \int \limits _\Omega \nabla \mbox{{$v$}} : \mbox{{$P$}}(\mbox{{$u$}}) \,\text{d}\Omega = 0 \,, \) 
